Interval Predictors for Unknown Dynamical Systems: an Assessment of Reliability

This paper presents new results for the assessment of reliability of predictive interval maps constructed using a consistency criterion with respect to a finite number of observations. Given a regression vector, our predictive map returns an interval in which the output of the unknown system is likely to fall at the next time instant. The key question we address is then the following: if the map has been constructed based on N observations, what is the probability that the next (unseen) output will actually fall in the interval predicted by the map? We answer to this fundamental question in two different settings. In the first setting we assume that the observations are statistically independent and identically distributed, while in the second setting we study the case when the observations are generated by a mixing remote process. This latter case is the most relevant in the applications, since it allows for statistical dependence between past and future observations.